Chapter 2
Models of Growth: Rates of Change





2.5 Modeling Population Growth

2.5.2 Solutions of the Differential Equation

In Chapter 1 we raised some questions for which the answers had to be expressed in terms of functions rather than in terms of numbers or variables — for example, “What functions are additive?” or “What is the inverse of base-\(10\) exponentiation?” Now we have posed another kind of problem that calls for an answer of the same type: Given an equation that describes an instantaneous rate of change, what functions satisfy that equation?

Suppose we set k = 0.25 . What does it mean for a function to satisfy the differential equation

d ⁢ P d ⁢ t = 0.25 P ?

Consider the function P 1 defined by

P 1 ( t ) = 3 e 0.25 t .

This function is a solution, because

d d ⁢ t P 1 ( t ) = 0.25 ⋅ 3 e 0.25 t = 0.25 P 1 ( t )

for all values of \(t\). On the other hand, the function P 2 defined by

P 2 ( t ) = t 2

does not satisfy the differential equation, since

d d ⁢ t P 2 ( t ) = 2 t

and

0.25 P 2 ( t ) = 0.25 t 2

do not define the same function.

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